Proof. For given \(\psi (\cdot )\in L_{ {\cal F}_T }^\infty (0,T;\mathbb {R}^n)\)  and \(y(\cdot )\in L_\mathbb {F}^\infty (0,T;\mathbb {R}^n)\) , it follows from Proposition REF  (or [1]}) that for \(\hbox{\rm a.e.}\,t\in [0,T]\) , the following BSDE
\(\eta (t,r)=\psi (t)+\int _r^T g(t,s,y(s),\zeta (t,s))ds-\int _r^T \zeta (t,s)dW(s),\ \ r\in [t,T]\) 
